FR0201128 Gestiors INIS )oc Enreg. la.; 4*TRN£ Mass formula for T = 0 and T = 1 ground states of N = Z nuclei E. Baldini-Neto1, C.L. Lima1 and P. Van Isacker2 institute de Fisica, Universidade de Sao Paulo C.P. 66318, 05315:970 Sao Paulo, Brazil ' 2GANIL, BP 55027, F-14076 Caen Cedex 5, France GANIL P 02 04 O O Mass formula for T = 0 and T = 1 ground states of N = Z nuclei E. Baldini-Neto and C.L. Lima Nuclear Theory and Elementary Particle Phenomenology Group, Instituto de Fisica, Universidade de Sao Paulo, C.P. 66318, 05315-970 Sao Paulo, Sao Paulo, Brazil P. Van Isacker Grand Accelerateur National d'lons Lourds, BP 55027, F-14076 Caen Cedex 5, France (January 17, 2002) Abstract An algebraic model is developed to calculate the T = 0 and T = 1 ground- state binding energies for N = Z nuclei. The method is tested in the sd shell and is then extended to the 28-50 shell which is currently the object of many experimental studies. I. INTRODUCTION The investigation of the behavior of nuclei under extreme conditions has become an impor- tant tool to reveal new facets of nuclear matter. In particular, nuclei at the frontiers of the valley of stability constitute nowadays the most active research area of nuclear structure physics. With the advent of new radioactive beam facilities it is now possible to produce exotic nuclei that may have occurred naturally in the interior of exploding supernovas [1]. In short, extremely proton- and neutron-rich nuclei are now within experimental reach. Of specific interest to the present paper are the considerable experimental efforts to study nuclei with roughly the same number of neutrons and protons, N « Z. On the theoretical side, the challenge is to investigate whether models, developed for— and using the phenomenology of—stable nuclei, can still be applied in these new, as yet uncharted regions and, if not, to propose new approaches to do so based on the data available up to now. One of the main open questions is the validity of the nuclear shell model with its traditional magic numbers and of the usual treatment of the residual interaction among the valence nucleons [2,3]. The nuclear mass is a property of quintessential importance as it directly determines the stability of a nucleus. There are several theoretical approaches that reproduce the systematics of masses of nuclei and it is worthwhile to mention here two of them. The Extended Thomas-Fermi plus Strutinsky Integral [4] (ETFSI) is a high-speed approximation to the Hartree-Fock (HF) method with pairing correlations taken into account through BCS theory. In earlier versions a Wigner term was not included and this has been claimed to be the reason for the systematic calculated underbinding by about 2 MeV for even-even N = Z nuclei [5]. This effect persists for N = Z odd-odd systems and for N = Z ± 1 odd-mass nuclei but with less prominence. The mass formula based on the Finite Range Droplet Model (FRDM) [6] starts from a sophisticated liquid drop mass formula to which microscopic corrections due to shell effects are added. Both approaches have comparable numbers of parameters (about 15) and make reliable predictions with impressive success. In the FRDM and also in a recent ETFSI calculation [7] a Wigner (correction) term is included that specifically deals with the peculiar behavior of binding energies of N « Z nuclei and has a cusp-like behavior for N = Z. This treatment is effective for known masses but, as the correction is entirely ad hoc, it has the drawback that an extrapolation to unknown nuclei can be dangerous. It is therefore of interest to develop models based on simple physical principles that can account for the behavior of nuclear masses at the N ~ Z line. Many models have been used over the past years to investigate the structure of heavier N fa Z. We mention in particular recent applications of the Hartee-Fock-Bogolyubov (HFB) method that includes proton-neutron pairing correlations [8]. This approach is tailor-made for the treatment of TV sa Z nuclei but has the drawback of the lack of particle-number projection. Shell-model calculations [9] are generally extremely successful in reproducing spectroscopic nuclear data but require large configuration space diagonalizations. In addi- tion, no reliable theoretical procedure exists for deriving the monopole part of the residual interaction which hinders a comparison with measured nuclear binding energies. An alge- braic approach [10], which has affinities with the one presented here, utilizes the concept of dynamical supersymmetry for the calculation of the binding energies in the sd shell but does not go beyond it. In this paper, the Interacting Boson Model (IBM) [11] in its isospin invariant version is applied to proton-rich N ~ Z nuclei. The goal is to obtain reliable estimates of binding energies of T = 0 and T = 1 ground states in self-conjugate (N = Z) nuclei based on the concept of dynamical symmetry. The mass formula proposed is relatively simple and contains terms with an intuitively understandable significance. A particular ingredient is its treatment of the competition between isoscalar and isovector pairing. II. AN IBM-4 MASS FORMULA The Interacting Boson Model in its original version (IBM-1) [12] successfully describes col- lective aspects of nuclei through the use of s and d bosons which are thought to approximate pairs of valence nucleons coupled to angular momentum 0 and 2. No distinction is made between neutron and proton bosons. Whenever the difference between the neutron and pro- ton fluids is thought to play a role, one is forced to use more elaborate versions of the IBM. The neutron-proton interacting boson model, or IBM-2, was introduced mainly to provide a microscopic foundation to the model [13]. It uses as building blocks s and d bosons con- structed from neutron-neutron (nn) and proton-proton (pp) pairs solely. In the third and fourth versions of IBM, IBM-3 and IBM-4, the isospin quantum number is introduced in a natural way. In IBM-3 the entire isospin triplet T = 1 is included, leading to nn, np, and pp pairs with Tz = +1,0, -1 [14]. The IBM-4 considers both T = 0 and T = 1 pairs; the T — \ bosons are assigned an intrinsic spin S — 0 while T = 0 bosons carry an intrinsic spin S = 1 [15]. A justification of this choice is that the two-particle isospin-spin combinations (TS) = (10) and (TS) — (01) are lowest in energy and that they give rise to an SU(4) algebra which is the boson equivalent of Wigner's supermultiplet algebra [16]. The mass region 28 < N & Z < 50 has a very rich structural behavior, presenting many aspects of nuclear motion. It is an ideal testing ground for various models since a proper description of the data relies on the interplay between T = 0 as well as T = 1 pairing and deformation-driving interactions. In addition, it is a region of intense experimental studies but with few experimental data available up to now. Very recently, the IBM-4 was applied to the spectroscopy of exotic N ~ Z nuclei in the pf5/299/2 shell [17]. In this approach the IBM-4 Hamiltonian is derived from a realistic shell- model Hamiltonian through a mapping carried out for A — 58 and 60 nuclei. The boson energies and the boson-boson interactions are thus derived microscopically and no parameter enters the calculation (since the shell-model interaction is considered as an input). This microscopically derived Hamiltonian gives good results in 62Ga (when compared to the shell model) and predicts the energy spectra of heavier N = Z nuclei (such 66As and 70Br). The approach is reasonably successful in obtaining a spectroscopy of low-spin states in N ~ Z nuclei. It makes use, however, of a complicated Hamiltonian and, moreover, calculations beyond 70Br seem difficult. Prompted by these considerations, in particular the need for reliable binding energy pre- dictions at the N = Z line and the existence of a microscopically derived IBM-4 Hamiltonian, we propose here a simple mass formula in the context of IBM-4. In previous work [18] one of us introduced an algebraic Hamiltonian (which can be regarded as the s-boson channel of the general IBM-4 Hamiltonian of Ref. [17]) with the specific aim to study the competition between the isovector and isoscalar pairing modes in self-conjugate nuclei. The s bosons do give rise only to the symmetric representation of UL(6) but this approximation can be justified in even-even and odd-odd N = Z nuclei (the only ones considered here) where this is indeed the favored U(6) representation [15]. These previous studies [17,18] suggest that the relevant terms in a simple IBM-4 mass formula must be taken from two different symmetry classifications: U(36) Z> (Ut(6) D • • O SOL(3)) f SUTs(4) ] • ^ UT5(6)D{ DSOT(3)®SOS(3) . (1) [ UT(3) ® U5(3) J ) Due to the overall symmetry of the boson state under U(36), the UL(6) and Uxs(6) repre- sentations have identical labels [JVj,..., N6\. The terms chosen from (1) for the calculation of binding energies of N — Z nuclei are the following. First, the linear and quadratic Casimir operators of UL(6) are included. The total number of bosons, N, labels the symmetric representations of Ux,(6) and, as a result, these Casimir operators take account of the bulk properties of the nucleus and lead to a smooth variation of the mass with particle number. The next two terms to be included are the quadratic Casimir operator of SUj-5(4) and the linear Casimir operator of Us(3).